The solution at large \(N\)  is standard, and proceeds by introducing the a field \(\xi = \vec{\phi }^{\, 2}\) , and a constraint
field, \(\epsilon \) ,
\({\cal L}_{\rm cons} = i \epsilon (\xi - \vec{\phi }^{\, 2})/2\) .
I only seek the solution for the symmetric phase, although the solution in the broken phase can also be determined
[1]}.
Using this constraint, the \(\vec{\phi }\)  and \(\xi \)  fields are integrated out to give the effective action
\({\cal S}_{\rm eff} = \frac{N}{2} {\rm tr} \log \Delta ^{-1}+ \int d^4 x ( \frac{ \epsilon ^2}{4 \lambda } + \frac{\widetilde{m}_\omega ^2 \omega _\mu ^2}{2} - g_\omega \omega _0 \rho _B ) ;\) 
